Remark 4.3 The one-dimensional analogue of c.density-cf, namely \(|\varphi _1|^c\)  being integrable for some \(c\ge 1\) , is equivalent to \(Z_n\)  having a bounded density \(f_{Z_N}\)  for some \(N\in \mathbb {N}\)  (or equivalently for all sufficiently large \(n\) ) by [1]}. This is necessary for any LLT for \(Z_n\)  to hold and therefore posed as a condition in all LLTs such as [1]}, [3]} and [4]}.

